"Model number";"Mechanism to ensure contiguity";"Algebraic formula";References
1;"An arc can direct from node i to neighbour node j only if the value assigned to i is less than the value assigned to j, so an arc directing away from a node will not return to it and therefore loops are prevented.";"(1)-(6);";"Miller et al. (1960); Duque et al. (2011)’s TreePRM model."
2;"Two trees are formed, one in primal graph, the other in dual graph, the interwoven structure of these two trees prevents the formation of loops.";"(1)-(6); (2)-(4) in Williams (2002).";"Williams (2002)"
3;"One selected site serves as a sink, every other selected site provides one unit of supply to the flow and all flows finally reach the sink.";"(1) and (2); (1)-(3) in Shirabe (2005).";"Shirabe (2005, 2009); Duque et al. (2011)’s FlowPRM model"
4;"The tail length of a selected site equals the number of arcs linking to it and that length is strictly larger than that of the site’s preceding sites.";"(1)-(2); (3)-(7) in Önal and Briers (2006).";"Önal and Briers (2006)"
5;"Solve a relaxed problem first with no contiguity constraints. If a cycle occurs, apply an inequality (a cut) to that cycle and solve again until there is no cycle in the solution.";"(1)-(6); (8) in Önal and Wang (2008).";"Önal and Wang (2008)"
6;"An amount of flow is injected into the partition from an outside node, each selected site consumes one unit of flow, residual flow is absorbed by a predefined variable.";"(1) and (2); (5)-(8) in Conrad et al. (2012).";"Conrad et al. (2012)"
7;"Capital flows from the supply node to trans-shipment nodes and for each selected site, capital can come into it from only one of its adjacent sites.";"(1) and (2); (2)-(7) in Jafari and Hearne (2013).";"Jafari and Hearne (2013); Jafari et al. (2017)"
8;"If a site is selected then at least one of its adjacent sites, which is nearer (structurally or functionally) to the centre, must also be selected.";"Minimize ;(2),(4),(6) and (8) in Önal et al. (2016).";"Önal et al. (2016); Wang and Önal (2016)"